mlsanityData to 14 Oct 2026

Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein space

arXiv 1910.05954 ↗First posted 14 Oct 2019stat.ML cs.LG cs.NAcited by 25

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Abstract (arXiv)

This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorithms on measure data. Our main result is that the embedding is (bi-)H"older continuous, when the reference density is uniform over a convex set, and can be equivalently phrased as a dimension-independent H"older-stability results for optimal transport maps.

Key facts. Cited by 25 papers in the map (data through 14 Oct 2026). First posted on arXiv 14 Oct 2019.

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Citing papers per year: 2021: 3; 2022: 5; 2023: 4; 2024: 5; 2025: 3; 2026: 5.

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  1. 2023Conditional Optimal Transport on Function Spacescited by 26
  2. 2022Distribution Regression with Sliced Wasserstein Kernelscited by 18
  3. 2022Supervised learning of sheared distributions using linearized optimal transportcited by 14
  4. 2023Linearized Wasserstein dimensionality reduction with approximation guaranteescited by 12
  5. 2021Quantitative Uniform Stability of the Iterative Proportional Fitting Procedurecited by 11
  6. 2022Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measurescited by 7
  7. 2021Learning to Generate Wasserstein Barycenterscited by 6
  8. 2023Fast Optimal Transport through Sliced Wasserstein Generalized Geodesicscited by 5
  9. 2022Tangential Wasserstein Projectionscited by 4
  10. 2024Resistance Distance and Linearized Optimal Transport on Graphscited by 2
  11. 2024Solving Prior Distribution Mismatch in Diffusion Models via Optimal Transportcited by 2
  12. 2025A primer on optimal transport for causal inference with observational datacited by 2
13 more
  1. 2021Convex Smoothed Autoencoder-Optimal Transport modelcited by 1
  2. 2024Point Cloud Classification via Deep Set Linearized Optimal Transportcited by 1
  3. 2024Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applicationscited by 1
  4. 2025Statistical Inference for Optimal Transport Maps: Recent Advances and Perspectivescited by 1
  5. 2026Rate-Optimal Noise Annealing in Semi-Dual Neural Optimal Transport: Tangential Identifiability, Off-Manifold Ambiguity, and Guaranteed Recoverycited by 1
  6. 2026Static and Dynamic Approaches to Computing Barycenters of Probability Measures on Graphscited by 1
  7. 2022Approximative Algorithms for Multi-Marginal Optimal Transport and Free-Support Wasserstein Barycenterscited by 0
  8. 2023On Wasserstein distances for affine transformations of random vectorscited by 0
  9. 2024Using Linearized Optimal Transport to Predict the Evolution of Stochastic Particle Systemscited by 0
  10. 2025Federated Flow Matchingcited by 0
  11. 2026Convergence of empirical subgradients for optimal transport-based objectivescited by 0
  12. 2026Near-Lipschitz stability of the Kim--Milman flow mapcited by 0
  13. 2026The Fundamental Limits of Valid Transport Map Estimationcited by 0